engleski

Triples which are D(n)-sets for several n's

For a nonzero integer n, a set of distinct nonzero integers {; ; ; ; ; ; ; ; a1, a2, ... , am}; ; ; ; ; ; ; ; such that aiaj + n is a perfect square for all 1 <= i < j <= m, is called a Diophantine m- tuple with the property D(n) or simply D(n)- set. D(1)-sets are known as simply Diophantine m- tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine m- tuple (i.e. D(1)-set) which is also a D(n)-set for some n not equal to 1. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also D(n)-sets for some n <> 1. However, the conjecture does not hold, since, for example, {; ; ; ; ; ; ; ; 8, 21, 55}; ; ; ; ; ; ; ; is a D(1) and D(4321)- triple, while {; ; ; ; ; ; ; ; 1, 8, 120}; ; ; ; ; ; ; ; is a D(1) and D(721)- triple. We present several infinite families of Diophantine triples {; ; ; ; ; ; ; ; a, b, c}; ; ; ; ; ; ; ; which are also D(n)- sets for two distinct n's with n <> 1, as well as some Diophantine triples which are also D(n)-sets for three distinct n's with n 6= 1. We further consider some related questions.

Diophantine m-tuples, elliptic curves

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